Question

How to prove that a statement is a tautology using logical equivalences?

Answer 1

To prove a statement is a tautology using logical equivalences, manipulate the statement to a tautological form using laws of logic and simplify it further using logical equivalences. For example, (P AND Q) OR (NOT P) can be proven to be a tautology.

Step-by-step explanation:

To prove that a statement is a tautology using logical equivalences, you can use laws of logic to manipulate the statement until it is in a recognizable tautological form. You can then further simplify the statement using logical equivalences to show that it is always true.

For example, let's say we want to prove the statement: (P AND Q) OR (NOT P) is a tautology.

1. Using the distributive law, we can expand the statement to: (P OR (NOT P)) AND (Q OR (NOT P)).
2. Using the law of negation, we can simplify (P OR (NOT P)) to TRUE.
3. Using the law of identity, we can simplify (Q OR (NOT P)) to (Q OR P).
4. Finally, using the distributive law in reverse, we can simplify TRUE AND (Q OR P) to (Q OR P).

Since (Q OR P) is always true, we have shown that the original statement is a tautology.